On two-quotient strong starters for ${\mathbb{F}}_q$

Abstract

Let $G$ be a finite additive abelian group of odd order $n$, and let $G^{\ast }=G\setminus \{0\}$ be the set of non-zero elements. A starter for $G$ is a set $S=\{\{x_i,y_i\}:i=1,\dots ,\frac{n-1}{2}\}$ such that $\{x_1,\dots ,x_{\frac{n-1}{2}},y_1,\dots ,y_{\frac{n-1}{2}}\}=G^{\ast }$ and $\{\pm (x_i-y_i):i=1,\dots ,\frac{n-1}{2}\}=G^{\ast }$. Moreover, if $\left|\left\{x_i+y_i:i=1,\dots ,\frac{n-1}{2}\right\}\right|=\frac{n-1}{2}$, then $S$ is called a strong starter for $G$. A starter $S$ for $G$ is a $k$ quotient starter if there exists $Q\subseteq G^{\ast }$ of cardinality $k$ such that $y_i/x_i\in Q$ or $x_i/y_i\in Q$, for $i=1,\dots ,\frac{n-1}{2}$. In this paper, we give examples of two-quotient strong starters for ${\mathbb{F}}_q$, where $q=2^{k}t+1$ is a prime power with $k>1$ a positive integer and $t$ an odd integer greater than 1.